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Variational Quantum Eigensolver
TQCI | November 14, 2019 | Bruyères-le-Châtel
Identification & validation of use-cases
Elvira SHISHENINA | Henri CALANDRA | Charles MOUSSA | Yuan
YAO
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Computing the ground state energy with VQE
A.Peruzzo et al. A variational eigenvalue solver on a quantum
processor (2014). arXiv:1304.3061
TQCI | November 14, 2019
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VQE | electronic Schrödinger equation
Finding the ground state energy = solving an eigenvalue
problem
intractable problem in general!
TQCI | November 14, 2019
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I. Fermionic Hamiltonian
II. Second quantization
III. Mapping from fermions to qubits – qubit
Hamiltonian(Jordan-Wigner or Bravyi-Kitaev transformation)
VQE | from fermions to qubits
,
TQCI | November 14, 2019
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VQE | from fermions to qubits• open source library for compiling
and analyzing quantum algorithms to simulate fermionic
systems, including quantum chemistry.
• plugins available for electronic structure computation (Psi4,
PySCF)
https://github.com/quantumlib/OpenFermion
TQCI | November 14, 2019
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VQE | design
CPU
TQCI | November 14, 2019
fermionic problem qubit Hamiltonian
prepare trial state
measure expectation value
calculate energy
adjust parameters
QPU
optim
ize
solution
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VQE | optimization loop
initial guess
ansatz qubit Hamiltonian
QPU
state preparation
… ……… …+
+
measurement
ENER
GY
CPU
QPU
TQCI | November 14, 2019
updated parameter
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VQE | designVQE library
TQCI | November 14, 2019
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VQE | benchmarking
TQCI | November 14, 2019
H2
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H2
VQE | benchmarking
TQCI | November 14, 2019
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ground state energies
LiH BeH H2O CH4 O2 CO2
VQE Total -7.75927010033 -14.4365668914 -72.9137048216
-38.9700162812 -144.2795640017 -179.930380621
VQE Atos -7.79160864682 -14.6002137511 -73.1510280024
-39.1240178446 -144.1520719862 -180.918374772
FCI OpenFermion -7.86613635619 -14.9569336087 -74.9908117277
-39.8060539918 -147.7434954566 -185.231532228
VQE | benchmarking
TQCI | November 14, 2019
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1 - 3 % error
VQE | benchmarking
ground state energies
LiH BeH H2O CH4 O2 CO2
VQE Total -7.75927010033 -14.4365668914 -72.9137048216
-38.9700162812 -144.2795640017 -179.930380621
VQE Atos -7.79160864682 -14.6002137511 -73.1510280024
-39.1240178446 -144.1520719862 -180.918374772
FCI OpenFermion -7.86613635619 -14.9569336087 -74.9908117277
-39.8060539918 -147.7434954566 -185.231532228
TQCI | November 14, 2019
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configuration
LiH BeH H2O CH4 O2 CO2
num_orbitals 6 6 7 9 10 15
num_qubit 12 12 14 18 20 30
VQE | benchmarking
TQCI | November 14, 2019
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VQE | freezing orbitals
TQCI | November 14, 2019
CO
O
1s
CCarbon
6
OOxygen
8
1s 2s 2p
1s 2s 2p
OOxygen
8 1s 2s 2p
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VQE | freezing orbitals
TQCI | November 14, 2019
CO
O
1s
CCarbon
6
OOxygen
8
1s 2s 2p
1s 2s 2p
OOxygen
8 1s 2s 2p
reduced from 30 to 24 qubits
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TQCI | November 14, 2019
chemical value chain
excess renewable energy
CO2
+H2 HCO2H
H2CO
H3COH
CH4
+H2
-H2O
+H2
+H2
-H2O
carbon dioxide
acids, esters
aldehydes alcohols, amineshydro-
carbons
ener
gy c
onte
nt
-179.9303
-186.2908
-112.4542
-113.5732
-38.9700
-72.9137
-72.9137
-1.1719
-1.1719
-1.1719
-1.1719
VQE | freezing orbitalsKlankermayer J, Leitner W. 2016
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VQE | noisy QPU
TQCI | November 14, 2019
1. Hardware model
2. Simulation method
H2 COBYLA noisy qpuCOBYLA
FCI OpenFermion
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VQE | further steps
• Investigating the scalability of VQE
• Benchmarking against Imaginary Time Evolution (ITE)
• Extending VQE in order to estimate the rest of excited
states
• Exploring more complex use-cases
TQCI | November 14, 2019
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TQCI | November 14, 2019
Fe protein MoFe protein
F cluster P cluster M cluster
B.Hoffman et al. Mechanism of nitrogen fixation by nitrogenase
(2014). dx.doi.org/10.1021/cr400641x
Iron sulfide clusters (Fe_S_) of different sizes
Molybdenum nitrogenase
Afterwards
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Fe protein MoFe protein
F cluster P cluster M cluster
TQCI | November 14, 2019
B.Hoffman et al. Mechanism of nitrogen fixation by nitrogenase
(2014). dx.doi.org/10.1021/cr400641x
Iron sulfide clusters (Fe_S_) of different sizes
Molybdenum nitrogenase
Simulating F cluster is at the limit of
classical computers! (IBM Research, 2018)
Afterwards
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TQCI | November 14, 2019
4Fe - 4S
4 x 26 + 4 x 16 = 168 electrons
~ 10^50 permutationsF cluster
Afterwards
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TQCI | November 14, 2019
Afterwards
4Fe - 4S
4 x 26 + 4 x 16 = 168 electrons
~ 10^50 permutations
~ 192 qubits( 3-5 years )
F cluster
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Thank You!
TQCI | November 14, 2019
